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Mirrors > Home > ILE Home > Th. List > nn0ge2m1nn | GIF version |
Description: If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 4-Jan-2020.) |
Ref | Expression |
---|---|
nn0ge2m1nn | ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 109 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 𝑁 ∈ ℕ0) | |
2 | 1red 7971 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℝ) | |
3 | 2re 8988 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
4 | 3 | a1i 9 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ) |
5 | nn0re 9184 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
6 | 2, 4, 5 | 3jca 1177 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (1 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑁 ∈ ℝ)) |
7 | 6 | adantr 276 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (1 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑁 ∈ ℝ)) |
8 | simpr 110 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 2 ≤ 𝑁) | |
9 | 1lt2 9087 | . . . . . . 7 ⊢ 1 < 2 | |
10 | 8, 9 | jctil 312 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (1 < 2 ∧ 2 ≤ 𝑁)) |
11 | ltleletr 8038 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ 2 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((1 < 2 ∧ 2 ≤ 𝑁) → 1 ≤ 𝑁)) | |
12 | 7, 10, 11 | sylc 62 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 1 ≤ 𝑁) |
13 | elnnnn0c 9220 | . . . . 5 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁)) | |
14 | 1, 12, 13 | sylanbrc 417 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → 𝑁 ∈ ℕ) |
15 | nn1m1nn 8936 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ∨ (𝑁 − 1) ∈ ℕ)) | |
16 | 14, 15 | syl 14 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 = 1 ∨ (𝑁 − 1) ∈ ℕ)) |
17 | 1re 7955 | . . . . . . . . . . 11 ⊢ 1 ∈ ℝ | |
18 | 3, 17 | lenlti 8057 | . . . . . . . . . 10 ⊢ (2 ≤ 1 ↔ ¬ 1 < 2) |
19 | 18 | biimpi 120 | . . . . . . . . 9 ⊢ (2 ≤ 1 → ¬ 1 < 2) |
20 | 9, 19 | mt2 640 | . . . . . . . 8 ⊢ ¬ 2 ≤ 1 |
21 | breq2 4007 | . . . . . . . 8 ⊢ (𝑁 = 1 → (2 ≤ 𝑁 ↔ 2 ≤ 1)) | |
22 | 20, 21 | mtbiri 675 | . . . . . . 7 ⊢ (𝑁 = 1 → ¬ 2 ≤ 𝑁) |
23 | 22 | pm2.21d 619 | . . . . . 6 ⊢ (𝑁 = 1 → (2 ≤ 𝑁 → (𝑁 − 1) ∈ ℕ)) |
24 | 23 | com12 30 | . . . . 5 ⊢ (2 ≤ 𝑁 → (𝑁 = 1 → (𝑁 − 1) ∈ ℕ)) |
25 | 24 | adantl 277 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 = 1 → (𝑁 − 1) ∈ ℕ)) |
26 | 25 | orim1d 787 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → ((𝑁 = 1 ∨ (𝑁 − 1) ∈ ℕ) → ((𝑁 − 1) ∈ ℕ ∨ (𝑁 − 1) ∈ ℕ))) |
27 | 16, 26 | mpd 13 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → ((𝑁 − 1) ∈ ℕ ∨ (𝑁 − 1) ∈ ℕ)) |
28 | oridm 757 | . 2 ⊢ (((𝑁 − 1) ∈ ℕ ∨ (𝑁 − 1) ∈ ℕ) ↔ (𝑁 − 1) ∈ ℕ) | |
29 | 27, 28 | sylib 122 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 708 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 class class class wbr 4003 (class class class)co 5874 ℝcr 7809 1c1 7811 < clt 7991 ≤ cle 7992 − cmin 8127 ℕcn 8918 2c2 8969 ℕ0cn0 9175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-inn 8919 df-2 8977 df-n0 9176 |
This theorem is referenced by: nn0ge2m1nn0 9236 |
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